Vectors
7.4
JMerrill, 2007
Revised 2009
Definitions
• Vectors are quantities that are described by
direction and magnitude (size).
• Example: A force is a vector because in order to
describe a force, you must specify the direction in
which it acts and its strength.
• Example: The velocity of an airplane is a vector
because velocity must be described by direction
and speed.
• The vector 0, 0 is the zero vector. It has no
direction.
Representation of Vectors
• The velocities of 3 airplanes, two of which are
heading northeast at 700 knots, are represented
by u and v
u
w ≠ u or v, why?
The direction is different.
v
We say u = v to
indicate both planes
have the same
velocity.
w
Magnitude
• The magnitude of vector v is represented by the
absolute value of v.
• In the previous example, |u| = 700, |v|= 700, and
|w| = 700. We know that |u|=|v|=|w|, but
u ≠ w, and v ≠ w, why?
•
The direction is not the same!
Addition of Vectors
v  AB
AB  BC  AC
AC is the vector sum
of AB  BC
A
10
B
5
C
Addition of Vectors
• Addition is commutative, so a + b = b + a
b
b+a
a
a
a+b
b
• The vector sum is called the resultant.
Vector Subtraction
• A negative vector has the same magnitude, but in
the opposite direction.
v
-v
• v + (-v) = 0
• v – w means v + (-w)
v
w
Multiples of Vectors
-v
v
-2v
2v
-3v
3v
You Do
• Let u =
• Find 3u + 2v
• Find ½ u + 4v
• Find u – 2v
and v =
Scalar Multiplication
• Real numbers are often referred to as scalars.
• When we multiply a vector by a scalar, we use the
same rules that we are familiar with:
k(v + w) = kv + kw
k(mv) = kmv
Component Form
• From the tail to the tip of
vector v, we see:
• A 2 unit change in the xdirection, and
• A -3 unit change in the ydirection.
• 2 and -3 are the components
of v.
• When we write v = 2, -3 ,
we are expressing v in
component form.
2
3
Component Form
• You can count the number of
spaces to get the component
form or, you can subtract
the coordinates.
(x2,y2)
AB  x2  x1 , y2  y1
(y2 – y1)
• IT IS ALWAYS B – A!
• The magnitude of vector AB
is found using the distance
formula:
AB  ( x2  x1 ) 2  ( y2  y1 ) 2
(x1,y1)
(x2 – x1)
Example
• Given A(4, 2) and B(9, -1), express AB in component
form. Find AB
AB  9  4, 1  2  5, 3
AB  5  ( 3)  34
2
2
Vector Operations with Coordinates
 Vector Addition
v + u =
a,b + c,d = a+c, b+d
 Vector Subtraction
v - u =
a,b - c,d = a-c, b-d
 Scalar Multiplication
 kv =
k a,b = ka, kb
Example
• If u = 1, 3 and v = 2,5 , find:
• u+v
1+2, -3+5 = 3, 2
• u–v
1 - 2, -3 - 5 = -1, -8
• 2u – 3v
2 1, -3 - 3 2, 5 = 2, -6 - 6, 15 = -4, -21
Drawing
• Draw a parallelogram if you have a force.
• Draw using tip-to-tail if you have a change of
course.
• ALWAYS, ALWAYS, ALWAYS make your drawing in
proportion.
• And remember, heading/bearing/compass
direction is always measured clockwise from
magnetic north!
Example
• A force of 20N (20 Newtons) is pulling an object
east and another force of 10N is pulling the object
in the compass direction of 150o. Find the
magnitude and direction of the resultant force.
• Let O = object (and make it at the origin for ease
of computation)
Example
1. Draw what you know
2. Draw the parallelogram
150o
20N
3. Draw the resultant—that’s
what you’re looking for!
x
10N
60o
x
4. Use Law of Sines/Cosines
to find magnitude and
direction. We will use only
one of the triangles.
Example
1. We know the obtuse angle =
120 degrees.
2. Let r = resultant. Use the
Law of Cosines to find r.
10N
120o
x
20N
90o + 19.1o = 109.1o
So the resultant force is
26.46N in a direction of 109.1o
r 2  102  202  2(10)(20) cos120o
r  26.46
3. Now find angle x:
sin x sin120o

10
26.46
sin x  0.3273
x  19.1o
You Do:
• An airplane has a velocity of 400mph southwest.
A 50mph wind is blowing from the west. Find the
resultant speed and direction of the plane.
• We’re changing direction, so use tip-to-tail.
• The plane’s resultant velocity is about 366mph on
a course of approximately 219.5 degrees